Learn German: Betreten Sie dieses Grundstück etwa unbefugt? - Are you actually entering this property unauthorized?

In this sentence, etwa adds a strong sense of suspicion or negative expectation. The speaker doesn't just want a simple yes or no answer; they are expressing shock or making a mild accusation. It translates roughly to 'Are you seriously...?' or 'Are you actually...?' implying that the speaker strongly suspects you are doing something wrong.

I usually translate 'to step onto' with a preposition. Why does dieses Grundstück not have one?

The verb betreten (to enter or step onto) is directly transitive, meaning it takes a direct object in the accusative case without needing a preposition. This is why it is simply dieses Grundstück and not preceded by auf. Turning a prepositional action into a direct transitive one is a very common feature of verbs starting with the inseparable prefix be-.

Betreten Sie looks exactly like a formal command. How can I tell it is a question?

You are right that formal commands also start with the verb followed by the formal pronoun Sie (like in Betreten Sie den Rasen nicht!). However, the modal particle etwa is the massive clue here, as its specific function is to express disbelief in questions. In spoken German, the rising intonation at the end of the sentence would also make the question format completely obvious.

What is the origin of the word unbefugt?